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MECHANICS OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 11 Energy Methods

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Energy Methods Strain Energy Strain Energy Density Elastic Strain Energy for Normal Stresses Strain Energy For Shearing Stresses Sample Problem 11.2 Strain Energy for a General State of Stress Impact Loading Example Example Design for Impact Loads Work and Energy Under a Single Load Deflection Under a Single Load Sample Problem 11.4 Work and Energy Under Several Loads Castigliano’s Theorem Deflections by Castigliano’s Theorem Sample Problem 11.5

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf A uniform rod is subjected to a slowly increasing load The elementary work done by the load P as the rod elongates by a small dx is which is equal to the area of width dx under the load- deformation diagram. The total work done by the load for a deformation x 1, which results in an increase of strain energy in the rod. In the case of a linear elastic deformation, Strain Energy

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Strain Energy Density To eliminate the effects of size, evaluate the strain- energy per unit volume, As the material is unloaded, the stress returns to zero but there is a permanent deformation. Only the strain energy represented by the triangular area is recovered. Remainder of the energy spent in deforming the material is dissipated as heat. The total strain energy density resulting from the deformation is equal to the area under the curve to

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Strain-Energy Density The strain energy density resulting from setting R is the modulus of toughness. The energy per unit volume required to cause the material to rupture is related to its ductility as well as its ultimate strength. If the stress remains within the proportional limit, The strain energy density resulting from setting Y is the modulus of resilience.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Elastic Strain Energy for Normal Stresses In an element with a nonuniform stress distribution, For values of u < u Y, i.e., below the proportional limit, Under axial loading, For a rod of uniform cross-section,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Elastic Strain Energy for Normal Stresses For a beam subjected to a bending load, Setting dV = dA dx, For an end-loaded cantilever beam,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Strain Energy For Shearing Stresses For a material subjected to plane shearing stresses, For values of xy within the proportional limit, The total strain energy is found from

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Strain Energy For Shearing Stresses For a shaft subjected to a torsional load, Setting dV = dA dx, In the case of a uniform shaft,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.2 a)Taking into account only the normal stresses due to bending, determine the strain energy of the beam for the loading shown. b)Evaluate the strain energy knowing that the beam is a W10x45, P = 40 kips, L = 12 ft, a = 3 ft, b = 9 ft, and E = 29x10 6 psi. SOLUTION: Determine the reactions at A and B from a free-body diagram of the complete beam. Integrate over the volume of the beam to find the strain energy. Apply the particular given conditions to evaluate the strain energy. Develop a diagram of the bending moment distribution.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.2 SOLUTION: Determine the reactions at A and B from a free-body diagram of the complete beam. Develop a diagram of the bending moment distribution.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.2 Integrate over the volume of the beam to find the strain energy.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Strain Energy for a General State of Stress Previously found strain energy due to uniaxial stress and plane shearing stress. For a general state of stress, With respect to the principal axes for an elastic, isotropic body, Basis for the maximum distortion energy failure criteria,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Impact Loading Consider a rod which is hit at its end with a body of mass m moving with a velocity v 0. Rod deforms under impact. Stresses reach a maximum value m and then disappear. To determine the maximum stress m -Assume that the kinetic energy is transferred entirely to the structure, -Assume that the stress-strain diagram obtained from a static test is also valid under impact loading. Maximum value of the strain energy, For the case of a uniform rod,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Example Body of mass m with velocity v 0 hits the end of the nonuniform rod BCD. Knowing that the diameter of the portion BC is twice the diameter of portion CD, determine the maximum value of the normal stress in the rod. SOLUTION: Due to the change in diameter, the normal stress distribution is nonuniform. Find the static load P m which produces the same strain energy as the impact. Evaluate the maximum stress resulting from the static load P m

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Example SOLUTION: Due to the change in diameter, the normal stress distribution is nonuniform. Find the static load P m which produces the same strain energy as the impact. Evaluate the maximum stress resulting from the static load P m

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Example A block of weight W is dropped from a height h onto the free end of the cantilever beam. Determine the maximum value of the stresses in the beam. SOLUTION: The normal stress varies linearly along the length of the beam as across a transverse section. Find the static load P m which produces the same strain energy as the impact. Evaluate the maximum stress resulting from the static load P m

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Example SOLUTION: The normal stress varies linearly along the length of the beam as across a transverse section. Find the static load P m which produces the same strain energy as the impact. For an end-loaded cantilever beam, Evaluate the maximum stress resulting from the static load P m

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Design for Impact Loads For the case of a uniform rod, For the case of the cantilever beam Maximum stress reduced by: uniformity of stress low modulus of elasticity with high yield strength high volume For the case of the nonuniform rod,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Work and Energy Under a Single Load Previously, we found the strain energy by integrating the energy density over the volume. For a uniform rod, Strain energy may also be found from the work of the single load P 1, For an elastic deformation, Knowing the relationship between force and displacement,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Work and Energy Under a Single Load Strain energy may be found from the work of other types of single concentrated loads. Transverse loadBending coupleTorsional couple

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Deflection Under a Single Load If the strain energy of a structure due to a single concentrated load is known, then the equality between the work of the load and energy may be used to find the deflection. From statics, From the given geometry, Strain energy of the structure, Equating work and strain energy,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.4 Members of the truss shown consist of sections of aluminum pipe with the cross-sectional areas indicated. Using E = 73 GPa, determine the vertical deflection of the point E caused by the load P. SOLUTION: Find the reactions at A and B from a free-body diagram of the entire truss. Apply the method of joints to determine the axial force in each member. Evaluate the strain energy of the truss due to the load P. Equate the strain energy to the work of P and solve for the displacement.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.4 SOLUTION: Find the reactions at A and B from a free-body diagram of the entire truss. Apply the method of joints to determine the axial force in each member.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.4 Evaluate the strain energy of the truss due to the load P. Equate the strain energy to the work by P and solve for the displacement.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Work and Energy Under Several Loads Deflections of an elastic beam subjected to two concentrated loads, Reversing the application sequence yields Strain energy expressions must be equivalent. It follows that (Maxwell’s reciprocal theorem). Compute the strain energy in the beam by evaluating the work done by slowly applying P 1 followed by P 2,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Castigliano’s Theorem Strain energy for any elastic structure subjected to two concentrated loads, Differentiating with respect to the loads, Castigliano’s theorem: For an elastic structure subjected to n loads, the deflection x j of the point of application of P j can be expressed as

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Deflections by Castigliano’s Theorem Application of Castigliano’s theorem is simplified if the differentiation with respect to the load P j is performed before the integration or summation to obtain the strain energy U. In the case of a beam, For a truss,

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.5 Members of the truss shown consist of sections of aluminum pipe with the cross-sectional areas indicated. Using E = 73 GPa, determine the vertical deflection of the joint C caused by the load P. Apply the method of joints to determine the axial force in each member due to Q. Combine with the results of Sample Problem 11.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads P and Q. Setting Q = 0, evaluate the derivative which is equivalent to the desired displacement at C. SOLUTION: For application of Castigliano’s theorem, introduce a dummy vertical load Q at C. Find the reactions at A and B due to the dummy load from a free-body diagram of the entire truss.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.5 SOLUTION: Find the reactions at A and B due to a dummy load Q at C from a free-body diagram of the entire truss. Apply the method of joints to determine the axial force in each member due to Q.

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© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS ThirdEdition Beer Johnston DeWolf Sample Problem 11.5 Combine with the results of Sample Problem 11.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads P and Q. Setting Q = 0, evaluate the derivative which is equivalent to the desired displacement at C.

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